# Continuous bordering of matrices and continuous matrix decompositions

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## Summary.

Let \({\cal M}_n\) be the set of all real \(n\times n\)-matrices of rank \(\ge n-1\). We prove that for \(n\ge 2\) there are no continuous vector fields \(l,r:{\cal M}_n\rightarrow {\Bbb R}^n\) such that the bordered matrix \(\) is regular for all \(A\in{\cal M}_n\). This result has some relevance for the numerical analysis of steady state bifurcation. As a by-product we show that there is no nonvanishing continuous vector field \(u:{\cal M}_n^{(n-1)}\rightarrow {\Bbb R}^n\) with \(Au(A)=0\) for all \(A\in{\cal M}_n^{(n-1)}\), where \({\cal M}_n^{(n-1)}\subset {\cal M}_n\) is the set of all matrices of rank deficiency one. This implies that there is no singular value decomposition of \(A\) depending continuously on \(A\) in any matrix set which contains \({\cal M}_n^{(n-1)}\). As another application we prove that in general there is no global analytic singular value decomposition for analytic matrix valued functions of more than one real variable.

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